Optimal. Leaf size=185 \[ \frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x}{24 c^7}-\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x^3}{72 c^5}-\frac {b \left (8 c^2 d-3 e\right ) e x^5}{120 c^3}-\frac {b e^2 x^7}{56 c}-\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) \text {ArcTan}(c x)}{24 c^8}+\frac {1}{4} d^2 x^4 (a+b \text {ArcTan}(c x))+\frac {1}{3} d e x^6 (a+b \text {ArcTan}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {ArcTan}(c x)) \]
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Rubi [A]
time = 0.12, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {272, 45, 5096,
1275, 209} \begin {gather*} \frac {1}{4} d^2 x^4 (a+b \text {ArcTan}(c x))+\frac {1}{3} d e x^6 (a+b \text {ArcTan}(c x))+\frac {1}{8} e^2 x^8 (a+b \text {ArcTan}(c x))-\frac {b \text {ArcTan}(c x) \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{24 c^8}-\frac {b e x^5 \left (8 c^2 d-3 e\right )}{120 c^3}+\frac {b x \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{24 c^7}-\frac {b x^3 \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )}{72 c^5}-\frac {b e^2 x^7}{56 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 209
Rule 272
Rule 1275
Rule 5096
Rubi steps
\begin {align*} \int x^3 \left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac {x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )}{24+24 c^2 x^2} \, dx\\ &=\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \left (-\frac {6 c^4 d^2-8 c^2 d e+3 e^2}{24 c^8}+\frac {\left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x^2}{24 c^6}+\frac {\left (8 c^2 d-3 e\right ) e x^4}{24 c^4}+\frac {e^2 x^6}{8 c^2}+\frac {6 c^4 d^2-8 c^2 d e+3 e^2}{c^8 \left (24+24 c^2 x^2\right )}\right ) \, dx\\ &=\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x}{24 c^7}-\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x^3}{72 c^5}-\frac {b \left (8 c^2 d-3 e\right ) e x^5}{120 c^3}-\frac {b e^2 x^7}{56 c}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )-\frac {\left (b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right )\right ) \int \frac {1}{24+24 c^2 x^2} \, dx}{c^7}\\ &=\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x}{24 c^7}-\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) x^3}{72 c^5}-\frac {b \left (8 c^2 d-3 e\right ) e x^5}{120 c^3}-\frac {b e^2 x^7}{56 c}-\frac {b \left (6 c^4 d^2-8 c^2 d e+3 e^2\right ) \tan ^{-1}(c x)}{24 c^8}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{3} d e x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{8} e^2 x^8 \left (a+b \tan ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 174, normalized size = 0.94 \begin {gather*} \frac {105 a c^8 x^4 \left (6 d^2+8 d e x^2+3 e^2 x^4\right )+b c x \left (315 e^2-105 c^2 e \left (8 d+e x^2\right )+7 c^4 \left (90 d^2+40 d e x^2+9 e^2 x^4\right )-3 c^6 \left (70 d^2 x^2+56 d e x^4+15 e^2 x^6\right )\right )+105 b \left (-6 c^4 d^2+8 c^2 d e-3 e^2+c^8 \left (6 d^2 x^4+8 d e x^6+3 e^2 x^8\right )\right ) \text {ArcTan}(c x)}{2520 c^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 219, normalized size = 1.18
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{4} d^{2} c^{8} x^{4}+\frac {1}{3} d \,c^{8} e \,x^{6}+\frac {1}{8} e^{2} c^{8} x^{8}\right )}{c^{4}}+\frac {b \arctan \left (c x \right ) d^{2} c^{4} x^{4}}{4}+\frac {b \,c^{4} \arctan \left (c x \right ) d e \,x^{6}}{3}+\frac {b \,c^{4} \arctan \left (c x \right ) e^{2} x^{8}}{8}-\frac {b \,d^{2} c^{3} x^{3}}{12}-\frac {b \,c^{3} d e \,x^{5}}{15}-\frac {b \,c^{3} e^{2} x^{7}}{56}+\frac {b c \,d^{2} x}{4}+\frac {b c d e \,x^{3}}{9}+\frac {b c \,e^{2} x^{5}}{40}-\frac {b d e x}{3 c}-\frac {b \,e^{2} x^{3}}{24 c}+\frac {b \,e^{2} x}{8 c^{3}}-\frac {b \,d^{2} \arctan \left (c x \right )}{4}+\frac {b d e \arctan \left (c x \right )}{3 c^{2}}-\frac {b \,e^{2} \arctan \left (c x \right )}{8 c^{4}}}{c^{4}}\) | \(219\) |
default | \(\frac {\frac {a \left (\frac {1}{4} d^{2} c^{8} x^{4}+\frac {1}{3} d \,c^{8} e \,x^{6}+\frac {1}{8} e^{2} c^{8} x^{8}\right )}{c^{4}}+\frac {b \arctan \left (c x \right ) d^{2} c^{4} x^{4}}{4}+\frac {b \,c^{4} \arctan \left (c x \right ) d e \,x^{6}}{3}+\frac {b \,c^{4} \arctan \left (c x \right ) e^{2} x^{8}}{8}-\frac {b \,d^{2} c^{3} x^{3}}{12}-\frac {b \,c^{3} d e \,x^{5}}{15}-\frac {b \,c^{3} e^{2} x^{7}}{56}+\frac {b c \,d^{2} x}{4}+\frac {b c d e \,x^{3}}{9}+\frac {b c \,e^{2} x^{5}}{40}-\frac {b d e x}{3 c}-\frac {b \,e^{2} x^{3}}{24 c}+\frac {b \,e^{2} x}{8 c^{3}}-\frac {b \,d^{2} \arctan \left (c x \right )}{4}+\frac {b d e \arctan \left (c x \right )}{3 c^{2}}-\frac {b \,e^{2} \arctan \left (c x \right )}{8 c^{4}}}{c^{4}}\) | \(219\) |
risch | \(\frac {i b \,e^{2} x^{8} \ln \left (-i c x +1\right )}{16}+\frac {i b \,d^{2} x^{4} \ln \left (-i c x +1\right )}{8}+\frac {i b d e \,x^{6} \ln \left (-i c x +1\right )}{6}+\frac {x^{8} e^{2} a}{8}-\frac {i b \left (3 e^{2} x^{8}+8 d e \,x^{6}+6 d^{2} x^{4}\right ) \ln \left (i c x +1\right )}{48}+\frac {x^{6} e d a}{3}-\frac {b \,e^{2} x^{7}}{56 c}+\frac {x^{4} d^{2} a}{4}-\frac {b d e \,x^{5}}{15 c}-\frac {b \,d^{2} x^{3}}{12 c}+\frac {b \,e^{2} x^{5}}{40 c^{3}}+\frac {b d e \,x^{3}}{9 c^{3}}+\frac {b \,d^{2} x}{4 c^{3}}-\frac {b \,e^{2} x^{3}}{24 c^{5}}-\frac {b \,d^{2} \arctan \left (c x \right )}{4 c^{4}}-\frac {b d e x}{3 c^{5}}+\frac {b d e \arctan \left (c x \right )}{3 c^{6}}+\frac {b \,e^{2} x}{8 c^{7}}-\frac {b \,e^{2} \arctan \left (c x \right )}{8 c^{8}}\) | \(254\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 184, normalized size = 0.99 \begin {gather*} \frac {1}{8} \, a x^{8} e^{2} + \frac {1}{3} \, a d x^{6} e + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b d^{2} + \frac {1}{45} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b d e + \frac {1}{840} \, {\left (105 \, x^{8} \arctan \left (c x\right ) - c {\left (\frac {15 \, c^{6} x^{7} - 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} - 105 \, x}{c^{8}} + \frac {105 \, \arctan \left (c x\right )}{c^{9}}\right )}\right )} b e^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.49, size = 193, normalized size = 1.04 \begin {gather*} \frac {630 \, a c^{8} d^{2} x^{4} - 210 \, b c^{7} d^{2} x^{3} + 630 \, b c^{5} d^{2} x + 105 \, {\left (6 \, b c^{8} d^{2} x^{4} - 6 \, b c^{4} d^{2} + 3 \, {\left (b c^{8} x^{8} - b\right )} e^{2} + 8 \, {\left (b c^{8} d x^{6} + b c^{2} d\right )} e\right )} \arctan \left (c x\right ) + 3 \, {\left (105 \, a c^{8} x^{8} - 15 \, b c^{7} x^{7} + 21 \, b c^{5} x^{5} - 35 \, b c^{3} x^{3} + 105 \, b c x\right )} e^{2} + 56 \, {\left (15 \, a c^{8} d x^{6} - 3 \, b c^{7} d x^{5} + 5 \, b c^{5} d x^{3} - 15 \, b c^{3} d x\right )} e}{2520 \, c^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.58, size = 260, normalized size = 1.41 \begin {gather*} \begin {cases} \frac {a d^{2} x^{4}}{4} + \frac {a d e x^{6}}{3} + \frac {a e^{2} x^{8}}{8} + \frac {b d^{2} x^{4} \operatorname {atan}{\left (c x \right )}}{4} + \frac {b d e x^{6} \operatorname {atan}{\left (c x \right )}}{3} + \frac {b e^{2} x^{8} \operatorname {atan}{\left (c x \right )}}{8} - \frac {b d^{2} x^{3}}{12 c} - \frac {b d e x^{5}}{15 c} - \frac {b e^{2} x^{7}}{56 c} + \frac {b d^{2} x}{4 c^{3}} + \frac {b d e x^{3}}{9 c^{3}} + \frac {b e^{2} x^{5}}{40 c^{3}} - \frac {b d^{2} \operatorname {atan}{\left (c x \right )}}{4 c^{4}} - \frac {b d e x}{3 c^{5}} - \frac {b e^{2} x^{3}}{24 c^{5}} + \frac {b d e \operatorname {atan}{\left (c x \right )}}{3 c^{6}} + \frac {b e^{2} x}{8 c^{7}} - \frac {b e^{2} \operatorname {atan}{\left (c x \right )}}{8 c^{8}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{2} x^{4}}{4} + \frac {d e x^{6}}{3} + \frac {e^{2} x^{8}}{8}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.54, size = 374, normalized size = 2.02 \begin {gather*} x^4\,\left (\frac {\frac {a\,e^2}{c^2}-\frac {a\,e\,\left (2\,d\,c^2+e\right )}{c^2}}{4\,c^2}+\frac {a\,d\,\left (d\,c^2+2\,e\right )}{4\,c^2}\right )-x^6\,\left (\frac {a\,e^2}{6\,c^2}-\frac {a\,e\,\left (2\,d\,c^2+e\right )}{6\,c^2}\right )+x^5\,\left (\frac {b\,e^2}{40\,c^3}-\frac {b\,d\,e}{15\,c}\right )+\mathrm {atan}\left (c\,x\right )\,\left (\frac {b\,d^2\,x^4}{4}+\frac {b\,d\,e\,x^6}{3}+\frac {b\,e^2\,x^8}{8}\right )-x^2\,\left (\frac {\frac {\frac {a\,e^2}{c^2}-\frac {a\,e\,\left (2\,d\,c^2+e\right )}{c^2}}{c^2}+\frac {a\,d\,\left (d\,c^2+2\,e\right )}{c^2}}{2\,c^2}-\frac {a\,d^2}{2\,c^2}\right )-x^3\,\left (\frac {\frac {b\,e^2}{8\,c^3}-\frac {b\,d\,e}{3\,c}}{3\,c^2}+\frac {b\,d^2}{12\,c}\right )+\frac {x\,\left (\frac {\frac {b\,e^2}{8\,c^3}-\frac {b\,d\,e}{3\,c}}{c^2}+\frac {b\,d^2}{4\,c}\right )}{c^2}+\frac {a\,e^2\,x^8}{8}-\frac {b\,\mathrm {atan}\left (\frac {b\,c\,x\,\left (6\,c^4\,d^2-8\,c^2\,d\,e+3\,e^2\right )}{6\,b\,c^4\,d^2-8\,b\,c^2\,d\,e+3\,b\,e^2}\right )\,\left (6\,c^4\,d^2-8\,c^2\,d\,e+3\,e^2\right )}{24\,c^8}-\frac {b\,e^2\,x^7}{56\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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